The derived $p$-length of a $p$-solvable group with bounded indices of Fitting $p$-subgroups in its normal closures

Abstract

UDC 512.542

Let $G$ be a $p$-soluble group. Then $G$ has a subnormal series whose factors are $p^{\prime}$-groups or abelian $p$-groups. The smallest number of abelian $p$-factors of all such subnormal series of~$G$ is called the derived $p$-length of $G.$ A subgroup $H$ of a group $G$ is called Fitting if $H\leq F (G) .$ A functional dependence of the estimate of the derived $p$-length of a $p$-soluble group on the value of the indexes of Fitting $p$-subgroups in its normal closures is established.